Introductory Functional Analysis with Applications

"Provides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis." INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS.......................................... 1 PREFACE................................................................................. 4 CONTENTS................................................................................ 8 NOTATIONS............................................................................... 12 METRIC SPACES........................................................................... 16 1.1 Metric Space.................................................................... 17 1.2 Further Examples of Metric Spaces............................................... 24 1.3 Open Set, Closed Set, Neighborhood.............................................. 33 1.4 Convergence, Cauchy Sequence, Completeness...................................... 40 1.5 Examples. Completeness Proofs................................................... 48 1.6 Completion of Metric Spaces..................................................... 56 NORMED SPACES. BANACH SPACES............................................................ 64 2.1 Vector Space.................................................................... 65 2.2 Normed Space. Banach Space...................................................... 73 2.3 Further Properties of Normed Spaces............................................. 82 2.4 Finite Dimensional Normed Spaces and Subspaces.................................. 87 2.5 Compactness and Finite Dimension................................................ 92 2.6 Linear Operators................................................................ 97 2.7 Bounded and Continuous Linear Operators.........................................106 2.8 Linear Functionals..............................................................119 2.9 Linear Operators and Functionals on Finite Dimensional Spaces...................127 2.10 Normed Spaces of Operators. Dual Space.........................................132 INNER PRODUCT SPACES. HILBERT SPACES....................................................142 3.1 Inner Product Space. Hilbert Space..............................................143 3.2 Further Properties of Inner Product Spaces......................................152 3.3 Orthogonal Complements and Direct Sums..........................................157 3.4 Orthonormal Sets and Sequences..................................................166 3.5 Series Related to Orthonormal Sequences and Sets................................175 3.6 Total Orthonormal Sets and Sequences............................................183 3.7 Legendre, Hermite and Laguerre Polynomials......................................191 3.8 Representation of Functionals on Hilbert Spaces.................................203 3.9 Hilbert-Adjoint Operator........................................................210 3.10 Self-Adjoint, Unitary and Normal Operators.....................................216 FUNDAMENTAL THEOREMS FOR NORMED AND BANACH SPACES.......................................224 4.1 Zorn's Lemma....................................................................225 4.2 Hahn-Banach Theorem.............................................................228 4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces.................233 4.4 Application to Bounded Linear Functionals on C[a, b]............................240 4.5 Adjoint Operator................................................................246 4.6 Reflexive Spaces................................................................254 4.7 Category Theorem. Uniform Boundedness Theorem...................................261 4.8 Strong and Weak Convergence.....................................................271 4.9 Convergence of Sequences of Operators and Functionals...........................278 4.10 Application to Summability of Sequences........................................285 4.11 Numerical Integration and Weak* Convergence....................................291 4.12 Open Mapping Theorem...........................................................300 4.13 Closed Linear Operators. Closed Graph Theorem..................................307 FURTHER APPLICATIONS: BANACH FIXED POINT THEOREM........................................314 5.1 Banach Fixed Point Theorem......................................................315 5.2 Application of Banach's Theorem to Linear Equations.............................322 5.3 Application of Banach's Theorem to Differential Equations.......................330 5.4 Application of Banach's Theorem to Integral Equations...........................334 FURTHER APPLICATIONS: APPROXIMATION THEORY..............................................342 6.1 Approximation in Normed Spaces..................................................342 6.2 Uniqueness, Strict Convexity....................................................345 6.3 Uniform Approximation...........................................................351 6.4 Chebyshev Polynomials...........................................................360 6.5 Approximation in Hilbert Space..................................................367 6.6 Splines.........................................................................372 SPECTRAL THEORY OF LINEAR OPERATORS IN NORMED SPACES....................................378 7.1 Spectral Theory in Finite Dimensional Normed Spaces.............................379 7.2 Basic Concepts..................................................................385 7.3 Spectral Properties of Bounded Linear Operators.................................390 7.4 Further Properties of Resolvent and Spectrum....................................395 7.5 Use of Complex Analysis IN Spectral Theory......................................401 7.6 Banach Algebras.................................................................409 7.7 Further Properties of Banach Algebras...........................................413 COMPACT LINEAR OPERATORS ON NORMED SPACES AND THEIR SPECTRUM............................420 8.1 Compact Linear Operators on Normed Spaces.......................................421 8.2 Further Properties of Compact Linear Operators..................................427 8.3 Spectral Properties of Compact Linear Operators on Normed Spaces................435 8.4 Further Spectral Properties of Compact Linear ()perators........................443 8.5 Operator Equations Involving Compact Linear Operators...........................451 8.6 Further Theorems of Fredholm Type...............................................457 8.7 Fredholm Alternative............................................................466 SPECTRAL THEORY OF BOUNDED SELF-ADJOINT LINEAR OPERATORS................................474 9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators....................475 9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators............480 9.3 Positive Operators..............................................................485 9.4 Square Roots of a Positive Operator.............................................491 9.5 Projection Operators............................................................495 9.6 Further Properties of Projections...............................................501 9.7 Spectral Family.................................................................507 9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator.......................512 9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators................520 9.10 Extension of the Spectral Theorem to Continuous Functions......................528 9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator....531 UNBOUNDED LINEAR OPERATORS IN HILBERT SPACE.............................................538 10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators.................539 10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators.........545 10.3 Closed Linear Operators and Closures...........................................550 10.4 Spectral Properties of Self-Adjoint. Linear Operators..........................556 10.5 Spectral Representation of Unitary Operators...................................561 10.6 Spectral Representation of Self-Adjoint Linear operators.......................571 10.7 Multiplication Operator and Differentiation Operator...........................577 UNBOUNDED LINEAR OPERATORS IN QUANTUM MECHANICS.........................................586 11.1 Basic Ideas. States, Observables, Position Operator............................587 11.2 Momentum Operator. Heisenberg Uncertainty Principle............................591 11.3 Time-Independent Schrodinger Equation..........................................598 11.4 Hamilton Operator..............................................................605 11.5 Time-Dependent SchrOdinger Equation............................................613 APPENDIX 1: SOME MATERIAL FOR REVIEW AND REFERENCE......................................624 A1.1 Sets...........................................................................624 A1.2 Mappings.......................................................................628 A1.3 Families.......................................................................632 A1.4 Equivalence Relations..........................................................633 A1.5 Compactness....................................................................633 A1.6 Supremum and Infimum...........................................................634 A1.7 Cauchy Convergence Criterion...................................................636 A1.8 Groups.........................................................................637 APPENDIX 2: ANSWERS TO ODD-NUMBERED PROBLEMS............................................638 Section 1.1.........................................................................638 Section 1.2.........................................................................638 Section 1.3.........................................................................639 Section 1.4.........................................................................640 Section 1.5.........................................................................640 Section 1.6.........................................................................642 Section 2.1.........................................................................642 Section 2.2.........................................................................642 Section 2.3.........................................................................643 Section 2.4.........................................................................644 Section 2.5.........................................................................644 Section 2.6.........................................................................645 Section 2.7.........................................................................645 Section 2.8.........................................................................647 Section 2.9.........................................................................647 Section 3.1.........................................................................648 Section 3.2.........................................................................649 Section 3.3.........................................................................650 Section 3.4.........................................................................650 Section 3.5.........................................................................651 Section 3.6.........................................................................652 Section 3.7.........................................................................652 Section 3.8.........................................................................653 Section 3.9.........................................................................654 Section 3.10........................................................................654 Section 4.1.........................................................................655 Section 4.2.........................................................................655 Section 4.3.........................................................................656 Section 4.5.........................................................................656 Section 4.6.........................................................................657 Section 4.7.........................................................................657 Section 4.8.........................................................................658 Section 4.9.........................................................................658 Section 4.10........................................................................658 Section 4.11........................................................................659 Section 4 .12.......................................................................659 Section 4.13........................................................................659 Section 5.1.........................................................................660 Section 5.2.........................................................................662 Section 5.3.........................................................................663 Section 5.4.........................................................................663 Section 6.2.........................................................................663 Section 6.3.........................................................................665 Section 6.4.........................................................................665 Section 6.5.........................................................................666 Section 6.6.........................................................................667 Section 7.1.........................................................................667 Section 7.2.........................................................................668 Section 7.3.........................................................................668 Section 7.4.........................................................................669 Section 7.5.........................................................................670 Section 7.6.........................................................................670 Section 7.7.........................................................................670 Section 8.1.........................................................................671 Section 8.2.........................................................................671 Section 8.3.........................................................................671 Section 8.4.........................................................................673 Section 8.6.........................................................................673 Section 8.7.........................................................................675 Section 9.1.........................................................................676 Section 9.2.........................................................................676 Section 9.3.........................................................................677 Section 9.4.........................................................................677 Section 9.5.........................................................................678 Section 9.6.........................................................................679 Section 9.8.........................................................................679 Section 9.9.........................................................................680 Section 9.11........................................................................681 Section 10.1........................................................................681 Section 10.2........................................................................682 Section 10.3........................................................................682 Section 10.4........................................................................683 Section 10.5........................................................................683 Section 10.6........................................................................684 Section 11.2........................................................................684 Section 11.3........................................................................686 Section 11.4........................................................................686 Section 11.5........................................................................688 APPENDIX 3: REFERENCES..................................................................690 INDEX...................................................................................696 KREYSZIG The Wiley Classics Library consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Emil Artin Geometnc Algebra R. W. Carter Simple Groups Of Lie Type Richard Courant Differential and Integrai Calculus. Volume I Richard Courant Differential and Integral Calculus. Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics. Volume II Harold M. S. Coxeter Introduction to Modern Geometry. Second Edition Charles W. Curtis, Irving Reiner Representation Theory of Finite Groups and Associative Algebras Nelson Dunford, Jacob T. Schwartz unear Operators. Part One. General Theory Nelson Dunford. Jacob T. Schwartz Linear Operators, Part Two. Spectral TheorySelf Adjant Operators in Hilbert Space Nelson Dunford, Jacob T. Schwartz Linear Operators. Part Three. Spectral Operators Peter Henrici Applied and Computational Complex Analysis. Volume IPower Senes-lntegrauon-Contormal Mapping-Locatvon of Zeros Peter Hilton, Yet-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin Kreyszig Introductory Functional Analysis with Applications P. M. Prenter Splines and Variational Methods C. L. Siegel Topics in Complex Function Theory. Volume I Elliptic Functions and Uniformizatton Theory C. L. Siegel Topics in Complex Function Theory. Volume II Automorphic and Abelian Integrals C. L. Siegel Topics In Complex Function Theory. Volume III Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry Erwin Kreyszig. Reprint. Originally Published: New York : Wiley, C1978. Bibliography: P. 675-679.